# Full text of "Treatise On Analysis Vol-Ii"

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```9   THE PLANCHEREL-GODEMENT THEOREM       371

A. Show that there exists a decomposition of H as a Hilbert sum of a subspace H0 and
a (finite or infinite) sequence of closed subspaces H* stable with respect to B, with the
following properties: (1) the restrictions to H0 of the operators belonging to B are all
zero; (2) each Hfc is the Hilbert sum of a finite sequence (Hfcp)i <p<rfc of subspaces of
the same dimension (finite or not) which are stable with respect to B; the restrictions
to Hfci of the operators belonging to B form the algebra of Hilbert-Schmidt operators
on Hki; moreover, for 2^p^ rk, there exists an isometric isomorphism Tp of HfcJ.
onto Hfcp such that, if Ui is the restriction to Hki of an operator U e B, then the
restriction of U to Hfcp is TpUiTp1.

9. THE PLANCHEREL-GODEMENT THEOREM

In this section we shall study commutative (but not necessarily complete)
Hilbert algebras. More generally it will be convenient, in view of applications,
to consider a commutative algebra A with involution, endowed with a bitrace g
(15.7) satisfying conditions (U) and (N) of (15.6). We shall denote by n^ the
two-sided ideal of elements s e A such that g(s, s) = 0, and by ng : A -*• A/ng
the canonical mapping. We recall that A/ng is canonically endowed with a
structure of Hilbert algebra (15.7). We shall assume throughout that the
prehilbert space A/ng is separable, and hence is a dense subspace of a separ-
able Hilbert space, which we denote by Hg. Thus, starting from g, we obtain
canonically a representation of A in H^ (15.6.10), which we denote by Ug.
The image of A under Ug is a commutative subalgebra with involution
(15.6.1) of £?(Hg). Let ^denote its closure in &(Hg), so thatsfgis a com-
mutative star algebra (and therefore consists of normal operators (15.4.11)).
We shall assume that this algebra jfg is separable (this is not a consequence
of the separability of A/ng, cf. Problem 1).

A particular example of a trace on A (which therefore gives rise to a
bitrace, by the canonical procedure (15.6.2)) is provided by the hermitian
characters of A, i.e., the characters % of A which satisfy

(15.9.1)                                        *(**) = Xto

for all x e A. It follows that x(x*jc) = I x(x)\2 ^ 0, and if we put

g(x, y) = i(y*x) = x(y)x(x)>
the condition (U) of (15.6) is trivially verified, because

g(st, st) = \X(s)\2 \x(t)\2 = \x(s)\2ff(t, t\

On the other hand, the ideal n^, which is here the kernel of #, is a hyperplane
in A, so that the algebra A/ng can be identified with C, and the condition (N)Banach
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